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Equation 30 · The Bend an Elevator Cannot Fake

What does this equation mean?

r:=[I−B(B⊤C−1B)+B⊤C−1]y,r := \left[I - B\left(B^\top C^{-1}B\right)^{+}B^\top C^{-1}\right] y,

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operations[I - B(B^top C^-1B)^+B^top C^-1] y
Result or conditionr :
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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rr

Symbol r

r is part of the quantity the equation computes from the expression on the right.

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II

Symbol I

I is one of the signed contributions combined to compute the quantity on the left.

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BB

Symbol B

B is one of the signed contributions combined to compute the quantity on the left.

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B⊤B^\top

Symbol B^top

BtB^top is one of the signed contributions combined to compute the quantity on the left.

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C−1C^{-1}

Symbol C^-1

C−C^-1 is one of the signed contributions combined to compute the quantity on the left.

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yy

Symbol y

y is one of the signed contributions combined to compute the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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subtraction

subtraction

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

A tempting shortcut suggests itself immediately: pick any closed loop of clocks and sum the edge readings around it. If the sum fails to return to zero, something has broken the idea that a single scalar potential — curved or otherwise — assigns each clock one honest number. Formally, define the fitted residual r:=[I−B(B⊤C−1B)+B⊤C−1]yr := \left[I - B\left(B^\top C^{-1}B\right)^{+}B^\top C^{-1}\right] y. the part of y that weighted least squares could not explain with any node potential at all. It is tempting to read a nonzero r as curvature declaring itself. That reading is wrong, and the reason is worth deriving rather than waving at.

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